Models of Statistical Relationship

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Abstract
The choice among measures of relationship has increasingly become a matter of the interpretation of their intermediate values. Interpretations are important, but a prior question is the statistic's implicit model of a relationship—what it considers a perfect relationship, and what it considers a null relationship. A family of models based on combinations of certain maximum- and null-value conditions is analyzed in this paper. The distinction between the models can be used to shed light on the stakes involved in the choice among dichotomous variable measures as well as that among familiar ordinal statistics.The models are ordered in terms of their leniency, and the coefficients based on each model are specified. An empirical analysis shows that the different measures are positively correlated, but those measures based on different models can differ sharply from one another. Statistics based on the same model covary regardless of differences in their interpretations. Since different models are intended to measure different concepts, multiple coefficients can allow investigators to examine their data in greater detail. Several political examples of the use of multiple models are provided.

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